The nonlinear orbit equations are investigated analytically and numerically for a constant-amplitude electromagnetic wave (ωs, ks) with ordinary-mode polarization propagating perpendicular to a uniform magnetic field B0 êz. Relativisitic electron dynamics are essential in determining the maximum orbit excursion when the incident wave frequency ωs is near the nth harmonic of the electron cyclotron frequency Ωc. Coupled nonlinear equations are obtained for the slow evolution of the amplitude A(т) and phase ø(t) of the perpendicular orbit when ωs ≈ nΩc. The dynamical equations are reduced to quadrature and simplified analytically. At exact resonance (ωs = nΩc), if relativistic effects are (incorrectly) neglected, the maximum orbit excursion Amax approaches the unacceptably large value determined from the first (non-zero) solution to Jn(ns ΩsAmax) = 0, which totally invalidates the non-relativistic approximation. (Here ns Ωs = cks/Ωc.) As a contrasting illustrative example, for n = 1, ωs = Ωc and moderate pump strength, weak relativistic effects limit the maximum orbit excursion to the approximate value , where єs = ns ωs (Vz/c) Vq/c ≪ 1, and Vq is the axial quiver velocity in the applied oscillatory field. Physically, relativistic effects produce a nonlinear frequency shift that dynamically ‘detunes’ the particle motion from exact cyclotron resonance, thereby limiting amplitude growth.

The effect of finite Larmor radius (FLR) on the stability of m = 1 small-axial-wavelength kinks in a z–pinch with purely poloidal magnetic field is investigated. We use the incompressible FLR MHD model; a collisionless fluid model that consistently includes the relevant FLR terms due to ion gyroviscosity, Hall effect and electron diamagnetism. With FLR terms absent, the Kadomtsev criterion of ideal MHD, 2r dp/dr + m2B2/μ0 ≥ 0 predicts instability for internal modes unless the current density is singular at the centre of the pinch. The same result is obtained in the present model, with FLR terms absent. When the FLR terms are included, a normal-mode analysis of the linearized equations yields the following results. Marginally unstable (ideal) modes are stabilized by gyroviscosity. The Hall term has a damping (but not absolutely stabilizing) effect – in agreement with earlier work. On specifying a constant current and particle density equilibrium, the effect of electron diamagnetism vanishes. For a z–pinch with parameters relevant to the EXTRAP experiment, the m = 1 modes are then fully stabilized over the crosssection for wavelengths λ/a ≤ 1, where a denotes the pinch radius. As a general z–pinch result a critical line-density limit Nmax = 5 × 1018 m–1 is found, above which gyroviscous stabilization near the plasma boundary becomes insufficient. This limit corresponds to about five Larmor radii along the pinch radius. The result holds for wavelengths close to, or smaller than, the pinch radius and for realistic equilibrium profiles. This limit is far below the required limit for a reactor with contained alpha particles, which is in excess of 1020 m–1.

Nonlinear waves, solutions of the Zakharov–Kuznetsov (ZK) equation for a dilute plasma immersed in a strong magnetic field, are studied numerically. It is found that the most unstable mode rides with the carrier wave and leads to period doubling. Owing to the simplicity of the ZK equation, this and other phenomena, similar to those observed for gravity waves on a water surface, can be fully investigated. Analytical methods are also explored. An expansion in the carrier-wave amplitude leads to good estimates for growth rates, a result so far unobtainable for water waves. This paper can be read independently of parts 1 and 2, but a brief summary of all three parts is given at the end.

We have considered the effect due to the presence of two-temperature electrons on the formation and propagation of cnoidal waves in a weak relativistic plasma. Since the cnoidal wave is equivalent to an infinite series of solitary waves, it is worthy of study for the situation where more than one solitary wave is important. The phase velocity of the modulated periodic wave is plotted as a function of the amplitude. The form of the wave for small values of ellipticity of the cnoidal wave shows up as a series of harmonics. Such cnoidal excitations are important in determining the anomalous transport coefficients of a plasma.

The previously obtained analytical solution to the Boltzmann equation for nonhomogeneous plasmas with relative large temperature and density gradients is generalized in the following sense. (i) The relatively simple Bhatnagar, Gross and Krook collision operator is replaced by the Fokker-Planck operator expressed in terms of relaxation rates (slowing down, energy exchange, etc.). (ii) The simple Lorentzian plasma model is replaced by a multi-component plasma model with realistic masses and temperatures.

The effect of parallel electron dynamics is incorporated in the study of the Varma mode, which is a flute-like electrostatic ion drift perturbation in an inhomogeneous magnetic field. Starting from the electron and ion continuity equations, an equation for the conservation of the ion magnetic moment, the generalized Ohm's law, and Faraday's and Ampère's laws, we derive a set of four nonlinearly coupled differential equations. The nonlinearities arising from the E × B0 convection of the density and magnetic-moment variations, nonlinear ion polarization drift and Lorentz force, and the coupling of the parallel electron flow to the perturbed magnetic field are included. In the linear limit we find that the Varma mode becomes coupled with the modified-convective-cell and Alfvén modes. The instability of the coupled system is analysed and an expression for the growth rate is obtained. In the nonlinear regime we present the localized dipole vortex solution of the coupled Varma and convective-cell modes as well as the coupled Varma and Alfvén modes. It is found that for the former case the nonlinear structure is a solution of a second-order partial differential equation, whereas for the coupled Varma and inertial Alfvén waves the dipole vortex belongs to a family of fourth-order differential equations. Our results can be useful in understanding the electrostatic and electromagnetic fluctuations in mirror reactors, tokamaks and astrophysical plasmas.

On the basis of the ideal MHD equations expressed in terms of Elsässer variables, a new set of equations has been derived that governs the dynamics of the inhomogeneous background plasma and the superimposed incompressible fluctuations. From these equations the dynamic equation for the two-point and two-time correlation tensor has been obtained, and subsequently the equations of motion for the various spectral densities related to energy, cross-helicity and residual energy or the Alfvén ratio have been established. This set of equations offers a new possibility of discussing and perhaps better understanding the mostly incompressible fluctuations observed in the solar-wind plasma and of analysing their radial evolution into interplanetary space and their spectral development. The scope of the paper is limited to giving mainly formal developments of the equations. A detailed evaluation of the many terms in the light of interplanetary observations is intended for the future, but is not presented in this paper.

The classical phenomenon of electron plasma oscillations has been investigated from new aspects. The applicability of standard normal-mode analysis of plasma perturbations has been judged from comparisons with exact numerical solutions to the linearized initial-value problem. We consider both Maxwellian and non-Maxwellian velocity distributions. Emphasis is on perturbations for which αλD is of order unity, where α is the wavenumber and λD the Debye distance. The corresponding large-Debye-distance (LDD) damping is found to substantially dominate over Landau damping. This limits the applicability of normal-mode analysis of non-Maxwellian distributions. The physics of LDD damping and its close connection to large-Larmor-radius (LLR) damping is discussed. A major discovery concerns perturbations of plasmas with non-Maxwellian, bump-in-tail, velocity distribution functions f0(ω). For sufficiently large αλD (of order unity) the plasma responds by damping perturbations that are initially unstable in the Landau sense, i.e. with phase velocities initially in the interval where df0/dw > 0. It is found that the plasma responds through shifting the phase velocity above the upper velocity limit of this interval. This is shown to be due to a resonance with the drifting electrons of the bump, and explains the Penrose criterion.

If the rotamak is regarded as a spherical field-reversed mirror then, according to conventional ideal MHD analysis, it should be unstable to co-interchange modes localized near the vortex point of the magnetic field. It is shown that to study these instabilities in a typical rotamak plasma, the Hall term in Ohm's law cannot be ignored. The effect of the Hall term on the ideal MHD analysis of co-interchange modes is investigated and a stability criterion is derived.

A multiple-scales adiabatic perturbation theory is presented describing the adiabatic dissipation of the solitary vortex-pair solutions of the Hasegawa-Mima equation. The vortex parameter transport equations are derived as solvability conditions for the asymptotic expansion and are identical with the transport equations previously derived by Aburdzhaniya et al. (1987) using an energy- and enstrophy-conservation balance procedure. The theoretical results are compared with high-resolution numerical simulations. Global properties such as the decay in the enstrophy and energy are accurately reproduced. Local properties such as the position of the centre of the vortex pair, decay of the extrema in the vorticity and stream-function fields, and the dilation of the vortex dipole are also in good agreement. In addition, time series of vorticity–stream-function scatter diagrams for the numerical simulations are presented to verify the adiabatic ansatz.

A gyrokinetic formalism using magnetic co-ordinates is used to derive self-consistent, nonlinear Maxwell–Vlasov equations that are suitable for particle simulation studies of finite-β tokamak microturbulence and its associated anomalous transport. The use of magnetic co-ordinates is an important feature of this work since it introduces the toroidal geometry naturally into our gyrokinetic formalism. The gyrokinetic formalism itself is based on the use of the action-variational Lie perturbation method of Cary & Littlejohn, and preserves the Hamiltonian structure of the original Maxwell-Vlasov system. Previous nonlinear gyrokinetic sets of equations suitable for particle simulation analysis have considered either electrostatic and shear-Alfvén perturbations in slab geometry or electrostatic perturbations in toroidal geometry. In this present work fully electromagnetic perturbations in toroidal geometry are considered.

It is shown that, in addition to magnetic helicity and cross-helicity, a modified form of kinetic helicity is also conserved in MHD fluids if the Hall effect is taken into account. The consequence of including this modified kinetic helicity as a conservation integral in a variation of the system energy is the emergence of an unconditionally MHD-stable solution that is realized in migma plasmas and may be expected to emerge as a self-organized configuration from an initially turbulent magnetofluid.

A longitudinal undulating electric field in the lasing electron beam direction (or the longitudinal direction) and an electromagnetic wave propagating in the longitudinal direction (laser wave) drive a current that cannot be in the longitudinal direction. Thus this current can emit an electromagnetic wave having the same wave vector and polarization vector as those of the incident electromagnetic wave. This process is called stimulated bremsstrahlung in the longitudinal electric wiggler. Because of recent strong interest in this free electron lasing mechanism, it is appropriate to correct errors in numerical factors in the gain formula and equations in our paper.